Problem 1 - Brief Applied Calculus

Question

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \sqrt{x^{2}-3 x+1} $$

Step-by-Step Solution

Answer

Let \( g(x) = x^2 - 3x + 1 \) and \( f(x) = \sqrt{x} \), then \( f(g(x)) = \sqrt{x^2 - 3x + 1} \).

Step 1: Identify Inner Function

Observe that inside the square root expression, there is a quadratic polynomial. We want to express this as the function that is inside another function. Thus, we set the inner function as \( g(x) = x^2 - 3x + 1 \).

Step 2: Identify Outer Function

With the inner function identified, the outer function must be the one that acts on the result of \( x^2 - 3x + 1 \). Since the overall expression is a square root, we set the outer function as \( f(x) = \sqrt{x} \).

Step 3: Verify the Composition

To ensure the functions are correct, compose them: \( f(g(x)) = f(x^2 - 3x + 1) = \sqrt{x^2 - 3x + 1} \). This matches the original function, verifying our solution.

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a type of polynomial where the highest degree is 2. This simply means it includes a square term. In mathematical terms, it can be written in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In this exercise, the quadratic polynomial is \( x^2 - 3x + 1 \). This polynomial consists of:

• \( x^2 \): the square term, which is elevated to the second power.
• \( -3x \): the linear term, involving \( x \) to the first power and modifies the slope.
• \( +1 \): the constant term, which can shift the graph up or down on the y-axis.

Quadratic polynomials often form parabolic graphs, which can open upwards or downwards depending on the sign of the leading coefficient \( a \). This quadratic polynomial is part of the larger composite function, essentially acting as the core input which the next function, in this case, the square root, will process.

Square Root Function

The square root function is a foundational mathematical function represented by \( \sqrt{x} \). It operates by finding the number which, when multiplied by itself, equals \( x \). In simpler terms, it "undoes" the squaring process. In our example, the square root function is denoted as \( f(x) = \sqrt{x} \).This function has several noteworthy properties:

• Its domain includes only non-negative numbers \( x \ge 0 \), which ensures valid, real number outputs.
• It always yields non-negative results since it represents the principal square root.

In this exercise, the square root function serves as the outer function, which applies the square root operation to the result of the inner quadratic function \( g(x) = x^2 - 3x + 1 \). This step ensures that the original composition \( \sqrt{x^2 - 3x + 1} \) is precisely recreated.

Inner Function and Outer Function

Function composition involves creating a new function by applying one function to the results of another. This is much like placing blocks within blocks. Here, you deal with two main parts: the inner function and the outer function.

• The **inner function** does the initial work. Like the foundation of a building, it sets everything up. In the exercise, \( g(x) = x^2 - 3x + 1 \) is the inner function, a quadratic polynomial.
• The **outer function** then uses the outcome of the inner function. It wraps the previous results in another operation. For this exercise, \( f(x) = \sqrt{x} \) is the outer function, executing the square root on whatever the inner function provides.

By combining these, function composition becomes an effective way to manage complex expressions. Together, \( f(g(x)) = \sqrt{x^2 - 3x + 1} \) precisely mirrors the given function, confirming the correct identification of both parts.